Geodesic
On the number of decomposable trees
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (2006).

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A tree is called $k$-decomposable if it has a spanning forest whose components are all of size $k$. Analogously, a tree is called $T$-decomposable for a fixed tree $T$ if it has a spanning forest whose components are all isomorphic to $T$. In this paper, we use a generating functions approach to derive exact and asymptotic results on the number of $k$-decomposable and $T$-decomposable trees from a so-called simply generated family of trees - we find that there is a surprisingly simple functional equation for the counting series of $k$-decomposable trees. In particular, we will study the limit case when $k$ goes to $\infty$. It turns out that the ratio of $k$-decomposable trees increases when $k$ becomes large. 
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     author = {Wagner, Stephan G.},
     title = {On the number of decomposable trees},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities},
     year = {2006},
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Wagner, Stephan G. On the number of decomposable trees. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (2006). doi : 10.46298/dmtcs.3497. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3497/

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