Test: Tree

Tree: 

A tree is a connected subgraph of a connected graph containing all the nodes of the graph but containing no loops, i.e., there is a unique path between every pair of nodes.

The number of closed paths in a tree of the graph is zero. Therefore is not true for tree and graph.

Twig: The branches of the tree are called twigs.

Link: Those branches of the graph which are not in the tree.

Co-tree: All the links of a tree together constitute complement of the tree and is called co-tree, in which the number of branches is equal to b - (n - 1)

Where b is the number of branches of the graph.

Number of twigs: t = n - 1

Number of links: L = b - t = b – n + 1

A red-black tree is a balanced binary search tree with the following properties:

• Every node is colored red or black.
• Every leaf is a NIL node and is colored black.
• If a node is red, then both its children are black.
• Every simple path from a node to a descendant leaf contains the same number of black nodes.

Concept:

Tree:

• A tree consists of nodes or vertices that store information and often are labeled by a number or a letter.
• A tree consists of directed edges or undirected edges or both.
• The number of subtrees of a node is called its degree.
• A leaf or a terminal node is a node of degree zero
• A node that is not a leaf is called an interior node or an internal node

The degree of a node is the number of its children. The degree of a tree is the maximum degree of any of its nodes.

# unlabelled binary trees with 1 node

= 1

# with 2 nodes : -

= 2

# with 3 nodes :-

# with 4 nodes :-

An undirected graph G which is connected and acyclic is termed as a tree. G contains no cycles and if any edge is added to G a simple cycle is formed.

A star tree of order n is a tree with as many leaves as possible or in other words a star tree is a tree that consists of a single internal vertex and n-1 leaves. However, an internal vertex is a vertex of degree at least 2.

Every tree element is called a node and the lines connecting the elements are called branches. A finite tree structure has a member that has no superior and is called the “root” Or root node. Nodes that have no child are called leaf nodes.

A graph is called a bipartite graph if and only if it contains no cycle of odd length. Every tree is a bipartite graph and a median graph.

The number of labeled trees of k number of vertices is k^n-2. Two labeled trees are isomorphic if their graphs are isomorphic and the corresponding points of the two trees have the same labels.

A directed acyclic graph is known as a polytree whose underlying undirected graph is a tree. In other words, a directed tree is a directed graph which would be tree if the directions on the edges were ignored.