# Binomial Trees

Binomial Trees are one of the type of trees that are defined **recursively**. A Binomial tree of order 0 is a single node and a binomial tree of order n has a root node whose children are roots of binomial trees of order n-1, n-2, n-3, n-4, ......3,2,1,0.

### Properties of Binomial Tree

- There are
**2**^{n }nodes in a binomial tree of order**n**where n is the order and degree of tree(Fig 1). - Deleting roots yield binomial trees
**B**_{n-1},B_{n-2},....0. - B
_{n }has**nodes**at depth d.

### Binomial Tree of height/order(n)=0

When height =0 then **single** node will be present in Binomial tree(Fig 2).

### Binomial Tree at height (n)=1

When n=1 then 2^{1 }=**2 **nodes will be present in the tree(Fig 3).

### Binomial Tree of order=2

When n=2 then 2^{2 }=** 4 nodes **will be present in the tree.The subtree is **binomially ** attached to the root node(Fig 4).

### Binomial Tree of order 3

If order of tree is 3,then 2^{3 }nodes are present in the Binomial tree.**The root is connected to subtrees of order 0**(green color)**,1**(red)** and 2**(black)** **(Fig 5)**.**

### B_{n }has nodes at depth d.

If we have a binomial tree of order n,then we can trace the number of nodes at any depth by .For e.g the no of nodes of binomial tree of order 4 at depth 2 will be** 6**.(Fig 6).In fig 7,number of nodes at level 2 is 6 and is shown by red highlighted area.

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