Leaf multiplicity in a Bienaym\'e-Galton-Watson tree

This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaym\'e-Galton-Watson tree with critical offspring distribution $\xi$, conditioned on the tree being of size $n$. In particular, we show that if $S_n$ is the maximum multiplicity in a conditional Bienaym\'e-Galton-Watson tree, then $S_n = \Omega(\log n)$ asymptotically in probability and under the further assumption that ${\bf E}\{2^\xi\}<\infty$, we have $S_n = O(\log n)$ asymptotically in probability as well. Explicit formulas are given for the constants in both bounds. We conclude by discussing links with an alternate definition of multiplicity that arises in the root-estimation problem.


Introduction
Equivalence between two distinct mathematical objects is a far-reaching concept in mathematics. When two structures are similar, one may define a relation under which they are regarded as one and the same. The term "multiplicity" is often used to indicate the extent to which an object is, in some sense, not structurally unique (or how often it is repeated in a suitably-defined multiset). Towards a concept of the multiplicity of a node in a tree, consider the small example depicted in Fig. 1

Definitions and notation
Consider a tree T rooted at a node u. For a node v in the tree, we let T v denote the subtree rooted at v. Let v and w be nodes in the tree and let v = v 1 , v 2 , . . . , v n = u and w = w 1 , w 2 , . . . , w m = u be the paths from v and w, respectively, to the root. We say that v and w are identical and write v ≡ w if the paths have the same length and T vj and T wj are isomorphic as rooted ordered trees for 1 ≤ j ≤ n. For example, in Fig. 1, the nodes x and y are identical in this sense, but different from node z. It is clear that ≡ defines an equivalence relation on the set of nodes in the tree, so we may now define the multiplicity σ(v) of a node v to be the size of the equivalence class [v] under the relation. The leaf multiplicity (or simply multiplicity, when no confusion can arise) S(T ) of a rooted tree T is the maximum value of σ(v), taken over all nodes v of T . The name "leaf multiplicity" is motivated by the fact that the function σ increases monotonically away from the root, so that S(T ) remains the same when the maximum is only computed over the set of leaves of T .
Note that ≡ is not the only structural equivalence relation one can define on the set of nodes in a tree, and thus σ is only one of many possible notions of leaf multiplicity. Towards the end of this paper, we will explore an alternate definition µ of multiplicity which extends well to free trees as well as rooted trees, and discuss the relationship between the two functions σ and µ.

The Bienaymé-Galton-Watson model
For a random variable ξ taking values in the nonnegative integers, a Bienaymé-Galton-Watson tree [2] is a rooted ordered tree in which every node has i children with probability p i = P{ξ = i}. We say that ξ is the offspring distribution of the tree.
Trees arising from this process are often called Galton-Watson trees, but we include the name of I.-J. Bienaymé [3], since his work predates (and is more mathematically precise than) the analysis of F. Galton and H. W. Watson [7]; see [9] for an extended account of the history of branching processes. We shall deal with critical Bienaymé-Galton-Watson trees; that is, trees whose offspring distributions satisfy E{ξ} = 1 and V{ξ} ∈ (0, ∞). This restriction on the variance ensures that p i = 1, so that the tree is finite almost surely. The Bienaymé-Galton-Watson trees that we shall study are conditioned Bienaymé-Galton-Watson trees T n . Such trees are conditioned on |T | = n, where |T | is the number of nodes in the tree.

Rényi entropy
It will be convenient to simplify our notation with some information-theoretic definitions. Letting p i = P{ξ = i}, for α > 1 we define the Rényi entropy of order α [15] (see also [8]) to be the value As α → 1, this approaches the binary (Shannon) entropy [16] Since ξ will be fixed throughout the paper, for brevity we will let H α = H α (ξ) and H = H(ξ). Fix an offspring distribution ξ with mean 1 and nonzero finite variance; let T n be a conditioned Bienaymé-Galton-Watson tree of size n with this offspring distribution. The leaf multiplicity S(T n ) of this tree is a random variable, and will be denoted S n . The main result of this paper gives bounds on S n that are obeyed asymptotically in probability. Theorem 1. Let ξ be an offspring distribution with E{ξ} = 1 and V{ξ} ∈ (0, ∞). If S n is the multiplicity of a conditioned Bienaymé-Galton-Watson tree of size n with offspring distribution ξ, then letting we have for all ǫ > 0, as n → ∞, and under the further assumption that E{2 ξ } < ∞, we have the upper bound as n → ∞, where H 2 is the Rényi entropy of order 2 of the random variable ξ.
This theorem will be proved as two separate lemmas in the next section.

Asymptotics of the leaf multiplicity
In this section we derive asymptotic upper and lower bounds on S n . Before we begin, we observe that if p max = max i≥0 p i and 1 < α < β < ∞, we have the inequalities and defining H ∞ = log 2 (1/p max ), we have the equivalent chain of inequalities In particular, because we have assumed that V{ξ} = 0, we have the strict inequality for all k > 2.
We will prove the upper bound and lower bound for S n as two separate lemmas.
Lemma 2. Let ξ be an offspring distribution with mean 1 and nonzero finite variance σ 2 . Suppose further that E{2 ξ } is finite. If S n is the multiplicity of a conditioned Bienaymé-Galton-Watson tree of size n with offspring distribution ξ, then P{S n > (1 + ǫ)2 log 2 n/H 2 } → 0 for all ǫ > 0, where H 2 is the Rényi entropy of order 2 of the random variable ξ.
Proof: For 1 ≤ i ≤ n, let ξ i denote the degree of the ith node in preorder in the tree T n . For all 1 ≤ t < n, the partial sum t i=1 ξ i > t − 1 and n i=1 ξ i = n − 1. We will concentrate on the least common ancestor of the nodes in the largest equivalence class of T n . This node, call it w, has the property that the nodes in the equivalence class belong to k ≥ 2 different subtrees rooted at the children of w. The node w has (random) degree D, which we will deal with by summing over all possible degrees d. Let A wk denote the collection of all subsets of size k of the children of w (naturally, this collection is empty if w has fewer than k children). For x > 0, a node w, integers 2 ≤ k ≤ d, and a set A ∈ A wk , we let E(x, w, k, A) be the event that all the nodes in A are identical and their subtree sizes are at least x/k. Now for integers s ≥ x/k, we let E ′ (x, k, d, s, A) be the event that a randomly chosen node w of the tree T n has degree d and the leftmost k children of w are identical, with subtrees of size s. We have, by the union bound, Supposing that w is the jth node in preorder, E ′ (x, k, d, s, A) is the event that ξ j = d, (ξ j+1 , . . . , ξ j+s ) forms a tree, and (ξ j+rs+1,...,j+rs+s ) = (ξ j+1 , . . . , ξ j+s ) for all 1 ≤ r < k. Let us say that an integer j is "good" if these conditions hold when addition on the indices is done modulo n. Clearly, there are more good j than j satisfying the above conditions. Let G be the event that an index j chosen uniformly at random from {1, . . . , n} is "good"; let B be the event that (ξ 2 , . . . , ξ s ) forms a tree and (ξ rs+2 , . . . , ξ (r+1)s+1 ) = (ξ j+1 , . . . , ξ j+s ) for all 1 ≤ r < k. By a rotational argument due to Dwass [5], so letting we have P G | (ξ 1 , . . . , ξ n ) forms a tree = p d P{B}R.
Letting λ = gcd{i : i ≥ 1, p i > 0}, a lemma of Kolchin [12] states that uniformly in y, As the o(1) term does not depend on y, we find that where the o(1) term depends only on n. Assuming that ks whenever ks + d ≤ n/2. We now compute a bound on P{B}. We have and therefore, by independence of the ξ i , We can now combine all of these bounds. Substituting everything into (10), we have Since the inequality (8) was strict, there exists 0 < θ < 1 such that (20) we have provided that E{2 ξ } < ∞. Setting we find that P{S n ≥ x} → 0 as n → ∞. The next lemma presents a lower bound for S n .
Lemma 3. Let ξ be an offspring distribution with mean 1 and nonzero finite variance σ 2 . If S n is the multiplicity of a conditioned Bienaymé-Galton-Watson tree of size n with offspring distribution ξ, then Proof: Consider a complete k-ary tree of height L. This tree has k L leaves and 1 + k + · · · + k L−1 = (k L −1)/(k−1) internal nodes, all of degree k. The probability that an unconditioned Bienaymé-Galton-Watson tree takes this shape is call this probability q. For any real number x, the statement S n < x implies that no node in the tree can have the given k-ary tree as a subtree for any k L ≥ x, as the multiplicity of the k-ary tree is k L . Fix k ≥ 2 for now, let L be the first integer for which k ≥ x, and let y = k L . Observe that y ≤ kx. Denote the size of the k-ary tree by z = y + (y − 1)/(k − 1). We now consider the indices 1, 1 + z, 1 + 2z, 1 + 3z, . . . in {1, . . . , n − z}. Let Y i be the event (and Y i c its complement) that (ξ i , . . . , ξ i+z−1 ) defines precisely the k-ary tree, where i is in the set of indices defined above, which has size ⌊(n − z)/z⌋. Note that By Dwass' cycle lemma [5], the probability that (ξ 1 , . . . , ξ n ) defines a tree is Θ(n 3/2 ), so Substituting (1 − ǫ) log 2 n/ log 2 (1/γ) for x, and noting that z = Θ(log n), we observe that this bound tends to 0.

The maximal leaf-degree
Let T n be a random critical Bienaymé-Galton-Watson tree of size n. We let ξ u be the degree of the node u and let λ u be the number of children of u that are leaves in T n , i.e., the leaf-degree of u. We denote by L n the random variable max u∈Tn λ u ; it is clear that the multiplicity S n satisfies M n ≥ L n . The next lemma shows that when the tail of the offspring distribution ξ decays at a rate slower than exponential, the ratio L n / log n → ∞ in probability. So while our condition in the upper bound that E{2 ξ } be finite might have seemed somewhat artificial at first glance, we essentially cannot do without it.
in probability along a subsequence, as n → ∞.
The proof of this lemma uses Kesten's limit tree [11] for the offspring distribution ξ, whose construction we briefly recall here (see also [13]). Kesten's infinite tree, denoted T ∞ , is obtained by iterating the following step. Let the root of T ∞ be marked. A marked node has ζ children, where P{ζ = i} = ip i and p i = P{ξ = i}. Observe that ζ ≥ 1 and E{ζ} = E{ξ 2 } = σ 2 + 1. Of these ζ children, a random child is marked and all others are unmarked. The unmarked nodes are roots of independent (unconditioned) Bienaymé-Galton-Watson trees. The procedure is then repeated for the sole marked node. Proof: We argue by coupling T n with T ∞ . Let (T n , k) and (T ∞ , k) denote the truncations of T n and T ∞ , respectively, to the nodes at distance ≤ k from the root. Then, denoting the total variation distance by TV, it is known that if the sequence (k n ) is o( √ n) (see, e.g., [10] and [17]). We couple (T n , k n ) and (T ∞ , k n ) such that To show that L n / log n → ∞ in probability, it suffices to show this for L ′ n , the maximal leaf-degree among all marked nodes of (T ∞ , k n ) at distance < k n from the root. Let ζ 0 , ζ 1 , . . . , ζ kn−1 be the degrees of the marked nodes in (T ∞ , k n ), indexed by their distance from the root, let λ i be the leaf-degree corresponding to ζ i . Now, fix a constant c and let A i be the event that λ i ≤ c log n; we have Setting k n = ⌈n 1/3 ⌉ + 1, we have Note that λ 0 ∼ binomial(ζ 0 − 1, p 0 ), so that P{λ 0 ≤ p 0 ζ 0 /2 | ζ 0 } ≤ 1/2 for ζ 0 large enough, by the law of large numbers. Therefore, for n large enough, we have To conclude the proof, we must show that n 1/3 P{ζ 0 > 2c log n/p 0 } → ∞ along a subsequence of n.
Note that if for every ρ < 1, p n > ρ n for all n large enough, then L n / log n → ∞ in probability (instead of just along a subsequence).

Examples
There exists an important link between certain offspring distributions of conditioned Bienaymé-Galton-Watson trees and families of "simply-generated trees" [14]. In this section we examine several important families of trees in the Bienaymé-Galton-Watson context, and give explicit asymptotic upper and lower bounds for the multiplicity. In each case, the two important parameters will be and We must also verify that E{2 ξ } is finite, if the upper bound is to hold. In particular, this latter condition always holds if ξ is bounded. A summary of this section is displayed in Table 1.

Full binary trees
These are trees in which every node must have exactly zero or two children, and arise from the distribution p 0 = p 2 = 1/2. We compute γ = 1/16 and H 2 = 1, so that asymptotically in probability. Because the multiplicity in a full binary tree must be a power of 2, in essence this means that there exists a sequence of integers (a n ) such that In other words, in general one cannot improve the ratio between the upper and lower bounds in Theorem 1 to a factor of less than 8 + ǫ.

Flajolet t-ary trees
Full binary trees are a special case of a Flajolet t-ary tree for t = 2 (see [6], p. 68). In general, these are trees whose non-leaf nodes each have t children, and they arise from the finite distribution p 0 = (t − 1)/t, p t = 1/t. We have so log 2 (1/γ) = log 2 e + log 2 t + o(1) as t → ∞. On the other hand, so H 2 ∼ 2 log 2 e/t as t → ∞. This means that as t gets large, the ratio between the upper and lower bound grows as t log t.

Catalan trees
When we set p 0 = p 2 = 1/4 and p 1 = 1/2, we obtain a family of trees often called Catalan trees, since the number of such trees on n nodes is 2n n /(n + 1). There is a one-to-one correspondence between Catalan trees on n nodes and full binary trees on 2n + 1 nodes, since one obtains a full binary tree from a Catalan tree by adding artificial external nodes to every empty slot, and this procedure is reversed by removing all leaves from a full binary tree. It is easy to see that the leaf multiplicity of a full binary tree is exactly double the multiplicity of its corresponding Catalan tree. By plugging in d = 2 above, the lower bound given by Lemma 3 is log 2 n/8, which makes sense since the correspondence with full binary trees tells us that the lower bound on the Catalan trees should be similar to log 2 (2n + 1)/8. We calculate H 2 = log 2 (8/3) and the upper bound is 2 log 2 n/ log 2 (8/3), so the ratio between the upper and lower bounds is 16/ log 2 (8/3).

Binomial trees
Catalan trees are a special case of a binomial tree. For integer parameter d ≥ 2, nodes in these trees have d "slots" that may or may not contain a child; so there are d i ways for a node to have i children, for 0 ≤ i ≤ d. These trees correspond to a binomial(d, 1/d) distribution. We compute (49) Note that taking the limit d → ∞, the binomial(d, 1/d) distributions approach a Poisson(1) distribution. Thus we see from our earlier discussion on the Cayley trees that H 2 = log 2 e 2 /(I 0 (2) + o d→∞ (1). This gives the respective lower and upper bounds log 2 n 2 − (4d − 2) log 2 (1 − 1/d) and 2 log 2 n log 2 e 2 /(I 0 (2) + o d→∞ (1) .
The lower bound tends to log 2 n/(2 + 4 log 2 e) as d → ∞, matching the lower bound we obtained for Cayley trees above.

Motzkin trees
Also known as unary-binary trees, these are trees in which each non-leaf node can have either one or two children. They correspond to the distribution with p 0 = p 1 = p 2 = 1/3. We easily compute γ = 1/81 and H 2 = log 2 3, which yields an asymptotic lower bound of log 2 n/(log 2 81) = log 81 n and an asymptotic upper bound of 2 log 2 n/ log 2 3 = 2 log 3 n. The ratio between the upper and lower bounds is 8.

Planted plane trees
These are trees with ordered children, so that each can be embedded in the plane in a unique way. They correspond to a geometric(1/2) distribution, with p i = 1/2 i+1 for all i. We find that γ = 1/256, so we have the asymptotic lower bound S n ≥ log 2 n/8. Unfortunately, we have E{2 ξ } = i≥0 1/2 = ∞, so Lemma 2 cannot be applied to give an upper bound here. We note that the maximal degree ∆ n of T n satisfies ∆ n / log 2 n → 1 in probability (see, e.g., [4], Lemma 6). However, this does not imply that S n = O(log n) in probability. Fig. 3: Different leaf multiplicities but the same automorphic leaf multiplicity.

Automorphic multiplicity
The multiplicity of a tree does not have a natural extension to unrooted trees, because whether or not two nodes are identical depends crucially on their position in relation to a distinguished root node u. In this section we briefly investigate an alternate notion of multiplicity that does extend nicely to free trees. It arises in the problem of root estimation in Galton-Watson trees described in [4]. We briefly recall some definitions. Let T be a rooted tree. By disregarding the parent-child directions of the edges, we obtain a free tree T F . Conversely, if we start with a free tree T F and any node u, we can define a rooting of T F at u to be the rooted tree T u obtained by fixing u as the root. This does not give rise to a unique tree in general, because children of a given node may hang on the wall in an arbitrary left-to-right order, but our new notion of multiplicity will treat all of these possible ordered trees the same. Let Aut(T F ) be the group of all graph automorphisms of T F , that is, bijections f from the set of vertices T F to itself such that for vertices u and v, f (u) is adjacent to f (v) whenever u is adjacent to v. We can then define an automorphism of T u to be a graph automorphism of T F such that the root u stays fixed. By a slight abuse of notation, we denote the set of these rooted-tree automorphisms by Aut(T u ); formally this is the stabilizer subgroup of Aut(T F ). We will say that two nodes v and w in T u are congruent and write v ∼ u w if v and w belong to the same orbit under the action of Aut(T u ). This means that there exists an element f of Aut(T u ) such that f (v) = w. It is clear that this gives us an equivalence relation on the set of all nodes of T u , and the automorphic multiplicity of a node v, denoted µ(u, v), is the size of the equivalence class of v under this relation. Since any node can be mapped to itself under an automorphism, µ(u, v) ≥ 1 for all v.
In fact, one can define the relation ∼ u , and consequently the function µ, purely in terms of the relation ≡. We have v ∼ u w if and only if there exists a permutation for every node in T u such that applying each permutation to the left-to-right ordering of its respective node's children results in a tree in which v ≡ w. The analogue of S in this setting is the automorphic (leaf) multiplicity M (T ) of a rooted tree T . If o is the root of the tree T , then M (T ) is the maximum value of µ(o, v) over all nodes v in the tree T . Fig. 3 illustrates the distinction between the automorphic and non-automorphic multiplicity. We have S(T 1 ) = M (T 1 ) = 4, since the two non-leaf children of the root have identical (and therefore congruent) subtrees. In T 2 , on the other hand, these subtrees are congruent but not identical, so that M (T 2 ) = 4 but the non-automorphic multiplicity of T 2 is only 2.
This definition is still somewhat at odds with the notion of multiplicity that arises in the root estimation problem from [4]. In that setting, one considers all graph automorphisms of the free tree, not just ones that fix the root. We will call the size of the orbit of a node under this larger action the free multiplicity µ F and if two nodes u and v are congruent under an arbitrary graph automorphism, then we write u ∼ F v and say that the two nodes are free-congruent. We also let M F (T ) denote the free (leaf) multiplicity, the maximum value of µ F over all nodes in the free tree T F . Fig. 4 shows the relation between the automorphic multiplicity of a rooted tree and the free multiplicity its free-tree counterpart. Note that M (T ) ≤ M F (T ) for any rooted tree T , since we have µ(u) ≤ µ F (u) for every node u. We shall spend the rest of this section showing that this inequality can more or less be reversed. First, we need three lemmas, and in their statements and proofs, we shall understand "multiplicity" to mean "free multiplicity". In the following proof, we also write [G : H] to denote the index of a subgroup H in a larger group G; that is, the cardinality of the coset space G/H. Proof: We may reduce to the case where one of u or v is a leaf. This is because if neither is a leaf, then it is not in the orbit of any leaf by graph automorphism, so we can remove all the leaves from the tree T without changing either of µ F (u) or µ F (v). This is done finitely many times since T is finite and always contains at least one leaf. Now without loss of generality, suppose u is the leaf and v is its unique neighbour. By the orbitstabilizer theorem, where stabilizers are taken with respect to the group Aut(T ). Every automorphism fixing u must permute its neighbours, but since u only has one neighbour, we have Stab(u) ⊆ Stab(v). Thus proving the lemma.
The next lemma formalizes the intuitive notation that in a free tree, the multiplicities are in some sense smaller towards the centre of the tree.
: : : : : : Then v cannot have strict maximal free multiplicity among the three nodes; that is, Proof: Suppose for contradiction that µ F (v) > µ F (u) and µ F (v) > µ F (w). Then, for each of the pairs of neighbours u − − v and v − − w, the multiplicity of one of the nodes must be an integer multiple of the multiplicity of the other, by the previous lemma. So there must be integers r, s > 1 such that The situation is illustrated in Fig. 5. Since µ F (v) = sµ F (u), u must have s−1 children in the orbit of v and thus have subtree rooted at each of these children be isomorphic to B. Similarly, since µ F (v) = rµ F (w), w must have r − 1 child subtrees isomorphic to A. We note that in order to satisfy the r, s > 1 requirements, we must have |A| ≥ (s − 1)|B| + 2 and |B| ≥ (r − 1)|A| + 2, where the additional +2 terms come respectively from nodes u and v (for |A|) or v and w (for |B|). This implies that |A| ≥ (s − 1)(r − 1)|A| + 2s, which is impossible if |A| ≥ 1 and r, s > 1. The contradiction tells us that v cannot have strict maximal multiplicity among the three nodes.
We have established that if we embed a free tree into the (x, y)-plane and then lift the nodes up by setting each node's z-coordinate to its multiplicity, then the result is a convex, spidery bowl or valley. This is illustrated in Fig. 6.
On a path between any two endpoints, the multiplicities decrease monotonically towards the centre of the tree before increasing monotonically towards the endpoint. There is a central connected core of nodes of minimal multiplicity and we are able to show that this minimal multiplicity cannot be greater than 2.
Lemma 7. If F = (V, E) is a finite free tree, then the node of minimal multiplicity in F has multiplicity Proof: The proof is by contraposition. Let u ∈ V (F ) be a node of minimal multiplicity and suppose µ F (u) > 2. Let C u be the orbit of u. There is a subtree F ′ whose endpoints are the members of C u ; since m > 2 and the graph is connected, there is necessarily at least one node v ∈ F ′ \ C u . By Lemma 6, we have µ F (v) ≤ µ F (u) but by minimality of µ F (u), we know that µ F (v) = µ F (u). So we can repeat the argument with C v to find that the tree is infinite (at each step we are removing µ F (u) nodes from the free tree, but the process never terminates).
Note that this argument does not work when µ F (u) = 2 because F ′ may simply consist of two nodes connected by one edge.
and this bound is the best possible.
Proof: Suppose first that n ≥ 3. Let v be a leaf of maximal automorphic multiplicity in T F , and let [v] denote the set of nodes that are free-congruent to v (so [v] = M F (T )). By Lemma 7, a node s of minimal automorphic multiplicity either has µ F (s) = 1 or µ F (s) = 2, and since we assumed that n ≥ 3, we can require that s not be a leaf. If µ F (s) = 1, then M (T s ) = M F (T ), since any automorphism of T F already fixes s. The nodes in [v] all lie in some subtrees of s, and without loss of generality, we may assume that they do not all lie in the same subtree, since if s ′ is the only child of s whose subtree contains nodes of [v], we can reroot the tree T s at s ′ instead without changing the maximum automorphic multiplicity. There are d ≥ 2 children of the root whose subtrees contain elements of [v]; each one contains an equal proportion of these nodes, so d divides M F (T ). If we reroot the tree at any node outside these subtrees, then the automorphic multiplicity of the tree does not change. If, on the other hand, we choose a node in one of these subtrees, then there are still (d − 1)M F (T )/d leaves that can still be shuffled amongst themselves, so the maximum automorphic multiplicity is (d − 1)M F (T )/d ≥ M F (T )/2.
If µ F (s) = 2, there is a node s ′ that is free-congruent to s, and there is mirror symmetry in the graph. This means that there is a way to split the graph along an edge such that the two sides have the exact same shape, one contains s, and the other contains s ′ . The side containing s has M F (T )/2 members of [v]; call this half [v] s and the other half [v] s ′ . When the tree is rooted at s, we find that M (T s ) = M F (T )/2, since any two members of [v] s can be exchanged and any two members of [v] s ′ can be exchanged (but exchanges cannot happen between the two subtrees). And rerooting the tree at an arbitrary node, it is clear that the automorphic multiplicity of the tree will not decrease.
When n = 1 the statement is trivial, and taking n = 2 shows that the bound is the best possible, because if T is the tree with a root and a single (leaf) child, then M F (T ) = 2 and M (T ) = 1.
This theorem tells us that the asymptotics of the free multiplicity are the same as the asymptotics of the automorphic multiplicity, up to a fudge factor of 2.
Because congruence of two nodes is immediately implied by their being identical under ≡, we have S(T ) ≤ M (T ) for all rooted trees T . Thus if M n = M (T n ) and F n = M F (T n ), where T n is a conditioned Galton-Watson tree of size n, then if γ is as defined in Lemma 3, the inequality holds with probability tending to 1.