Average number of full nodes in rooted m-ary tree

Question

I am looking for a formula to express the average number of full nodes (i.e. nodes having exactly $m$ children) in a $m$-ary tree having $n$ nodes, i.e., $$ \mu_{n}^{(m)} = \frac{\# \text{full nodes in all $m$-ary trees having $n$ nodes} }{\# \text{nodes in all $m$-ary trees having $n$ nodes}}$$.

As to the denominator, according to this answer, it sholud be given by $$n C_m^n = \binom{mn}{n} \frac{n}{(m-1)n + 1}$$ As to numerator, at the first glance, I tought that this answer would have provided a hint; but then I have realized that it is not the case because it gives how many trees have $n$ full nodes, but I need to know how many full nodes are in the set of all the $m$-ary trees having $n$ nodes.

Could someone help me to understand how to solve it and provide (if any) a usable formula for $\mu_{n}^{(m)}$?

Answer 1

Every $m$-ary tree is a node together with up to $m$ children, which are also $m$-ary trees. Let $T(x,y)$ be the generating function in which the coefficient of $x^ny^k$ is the number of $m$-ary trees with $n$ nodes and $k$ full nodes. Then $$ T(x,y) = x(1 + T(x,y) + \cdots + T(x,y)^{m-1} + yT(x,y)^m). $$ The generating function of the total number of $m$-ary trees is $A(x) = T(x,1)$, and the generating function of the total number of full nodes in $m$-ary trees is $B(x) = \frac{\partial}{\partial y} T(x,y)|_{y=1}$.

The generating function $A(x)$ satisfies the equation $$ A(x) = x(1 + A(x) + \cdots + A(x)^m). $$ The generating function $B(x)$ satisfies the equation $$ B(x) = x(B(x) + 2A(x)B(x) + 3A(x)^2B(x) + \cdots + mA(x)^{m-1}B(x) + A(x)^m), $$ and so $$ B(x) = \frac{xA(x)^m}{1-x(1+2A(x)+3A(x)^2+\cdots+mA(x)^{m-1})}. $$

You can try solving these equations.

For example, in the trivial case $m=1$, we have $A(x) = x(1 + A(x))$, and so $$ A(x) = \frac{x}{1-x} = \sum_{n \geq 1} x^n. $$ It follows that $$ B(x) = \frac{x\cdot \frac{x}{1-x}}{1-x} = \frac{x^2}{(1-x)^2} = \sum_{n \geq 1} (n-1)x^n. $$ Therefore the average number of full nodes in $1$-ary trees with $n$ nodes is $\frac{n-1}{1} = n-1$.