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The result I'm referring to is: the number of spanning trees in a graph $G$ is the same as the number of $G$-parking functions. … There is a bijective proof of this fact in "A family of bijections between G-parking functions and spanning trees" by D. Chebikin and P. Pylyavskyy. …

The particular case of the square of a tree is easy to handle by producing a greedy $(\Delta+1)$-coloring starting from a root vertex and extending. However, much stronger results are known: The $k$-t …

and $H$ is the generating function for rooted unlabelled trees. … For Q2, homeomorphism classes of trees on $n$ vertices correspond to homeomorphically irreducible trees (also sometimes called series-reduced tress, or topological trees, see OEIS) of at most $n$ vertices …

While $n^{n-2}$ counts the number of vertex labeled trees on $n$ vertices, the expression $2^n(n+1)^{n-2}$ counts the number of edge labeled trees on $n$ edges. … There is a bijection between edge labeled trees on $n$ vertices and proper $(n-1)$-dimensional polycubes of size $n$. …

We will denote by $B_n$ the set of increasing 0-1-2 trees on $n$ vertices. … From here we will make use of the weighted Cayley formula for labeled trees on a set $S={1,2,\dots,p}$, which can be obtained from the matrix tree theorem. …