凯莱树公式的一个简短证明
A short proof of Cayley's tree formula
摘要：
我们给出了一个简短的证明凯莱的树计算公式的数量不同的标记树的$n$顶点。由组合论元$$ T_n = \frac{n}{2} \sum_{k=0}^{n-2} \left ( \begin {array} {c} n-2 \\ k \end {array} \right ) T_{k+1} T_{n-k-1}; \ \ for \ n > 1 \ and \ T_1 = 1, $$导出$n$顶点上标号树个数的非线性递归关系，并证明了$T_n = n^{n-2}$，从而又一次证明了著名的Cayley树公式.
英文摘要：
We give a short proof of Cayley's tree formula for counting the number of different labeled trees on $n$ vertices. The following nonlinear recursive relation for the number of labeled trees on $n$ vertices is deduced from a combinatorial argument, $$ T_n = \frac{n}{2} \sum_{k=0}^{n-2} \left ( \begin {array} {c} n-2 \\ k \end {array} \right ) T_{k+1} T_{n-k-1}; \ \ for \ n > 1 \ and \ T_1 = 1, $$ and then it is proved that $T_n = n^{n-2}$, which gives yet another proof of the celebrated Cayley's tree formula.
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