Even searching for " ['number of trees' leaves](https://oeis.org/search?q=%22number+of+trees%22+leaves) " didn't reveal what I am looking for: an approach for calculating the (approximate) number of trees with exactly n nodes and m leaves. Any hints from MO?



Number of trees with n nodes and m leaves
The answer to this (very natural) question depends on your notion of "tree" (e.g. free, rooted) and the equivalence relation you employ (e.g. labelled, unlabelled). I haven't gone into the nitty-gritty details of all these results, but here's what I've found so far. There's likely published results I haven't found yet, but hopefully this helps to get you started.


We can compute $T\_{m,n}$, the number of non-isomorphic free trees with $m$ leaves and $n$ vertices, for small $m$ and large $m$. For example, (a) $T\_{3,n}$ is the number of partitions of $n-1$ into $3$ positive integer parts ([Sloane's A001399](https://oeis.org/A001399)), (b) $T\_{n-2,n}=\lfloor (n-2)/2 \rfloor$ and (c) $T\_{n-3,n}=\sum\_{j=0}^{n-5} \lfloor (n-3-j)/2 \rfloor$. The first result can be observed by deleting the vertex of degree 3 and the last two can be observed by colouring each non-leaf vertex by the number of adjacent leaves, then deleting the leaves.


Yu (8) seems to have given an algorithm for generating rooted trees with $m$ leaves. Wang (6) and Liu (3,4) considered the number of "structurally different" trees with $m$ leaves (according to MathSciNet). Bergeron, Labelle and Leroux (1) consider the expected number of leaves in trees that admit a certain automorphism. Lam (2) discusses embeddings of trees with $m$ leaves and discusses trees with $(d+1)d^{r+1}$ leaves for integers d and r.


Wilf (7. p. 163) gave a generating function for $\sum\_k T\_{k,n}^{\text{lab}}$ where $T\_{k,n}^{\text{lab}}$ is the number of labelled free trees with $m$ leaves and $n$ vertices. He also gives a formula for the average number of leaves in a labelled tree with $n$ vertices.


There is also this: K. Yamanaka, Y. Otachi, S.-I. Nakano [Efficient Enumeration of Ordered Trees with k Leaves](https://doi.org/10.1007/978-3-642-00202-1_13), which I haven't looked at yet.


(1) F. Bergeron, G. Labelle, and P. Leroux, [Computation of the expected number of leaves in a tree having a given automorphism, and related topics](https://doi.org/10.1016/0166-218X(91)90078-B), Discrete Appl. Math., 34 (1991), pp. 49-66.


(2) P. C. B. Lam, [On number of leaves and bandwidth of trees](https://doi.org/10.1007/BF02677426), Acta Math. Appl. Sinica (English Ser.), 14 (1998), pp. 193-196.


(3) B. L. Liu, The enumeration of directed trees with a given number of leaves and the enumeration of free trees, Kexue Tongbao, 32 (1987), pp. 244-247. In Chinese.


(4) B. L. Liu, Enumeration of oriented trees and free trees with a given number of leaves, Kexue Tongbao (English Ed.), 33 (1988), pp. 1577-1581.


(5) Q. Q. Nong, The degree sequence and number of leaves in a tree, J. Yunnan Univ. Nat. Sci., 24 (2002), pp. 167-171. In Chinese.


(6) Z. Y. Wang, An enumeration problem on ordered trees, J. Math. (Wuhan), 6 (1986), pp. 201-208.


(7) H. C. Wilf, Generatingfunctionology, Academic Press, 1990.


(8) Q. L. Yu, An algorithm for lexicographically generating ordered rooted trees with constraints on the number of leaves, Chinese J. Oper. Res., 6 (1987), pp. 71-72


