# Number of edges binary tree with n leaves contain

A binary tree consists of a finite set of nodes that is either empty, or consists of one specially designated node called the root of the binary tree, and the elements of two disjoint binary trees called the number of edges binary tree with n leaves contain subtree and right subtree of the root. Note that the definition above is recursive: This is appropriate since recursion is an innate characteristic of tree structures.

Tree terminology is generally derived from the terminology of family trees specifically, the type of family tree called a lineal chart. If every non-leaf node in a binary tree has nonempty left and right subtrees, the tree is termed a strictly binary tree.

Or, to put it another way, all number of edges binary tree with n leaves contain the nodes in a strictly binary tree are of degree zero or two, never degree one. A complete binary tree of depth d is the strictly binary tree all of whose leaves are at level d.

Since all leaves in such a tree are at level dthe tree contains 2 d leaves and, therefore, 2 d - 1 internal nodes. An almost complete binary tree with N leaves that is not strictly binary has 2 N nodes. There are two distinct almost complete binary trees with N leaves, one of which is strictly binary and one of which is not. There is only a single almost complete binary tree with N nodes.

This tree is strictly binary if and only if N is odd. For a complete or almost complete binary tree, storing the binary tree as an array may be a good choice. One way to do this is to store the root of the tree in the first element of the array. For example, the almost complete binary tree shown in Diagram 2 can be stored in an array like so:. However, if this scheme is used to store a binary tree that is not complete or almost complete, we can end up with a great deal of wasted space in the array.

If a binary tree is not complete or almost complete, a better choice for storing it is to use a linked representation similar to the linked list structures covered earlier in the semester:. Each tree node has two pointers usually named left and right. The tree class has a pointer to the root node of the tree labeled root in the diagram above.

Any pointer in the tree structure that does not point to a node will normally contain the value NULL.

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