Binomial tree

table of Contents

One, binomial tree

Second, the generating function of the binomial tree

Three, the number of nodes in the binomial tree

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One, binomial tree

Binomial tree is a set of fixed recursively defined trees:

B0 is a single-node tree,

Bn is an n-ary tree, the root node has n children, namely B0, B1...B n-1

 

Second, the generating function of the binomial tree

For Bn, its depth is n, and we define its generating function:

  , Where si is the number of nodes in the i-th layer

According to the definition of the binomial tree, we can get the recurrence of the generating function:

According to this recurrence formula, it can be found,

Therefore, the generating function of the binomial tree Tn is binomial (1+x)^n

 

Three, the number of nodes in the binomial tree

The number of nodes in the i-th layer of the binomial tree is the binomial coefficient C(n, i)

The total number of nodes in the binomial tree is 2^n