# Concept of Tree

*For a given connected graph of a network, a connected subgraph is known as a tree of the graph if the subgraph has all the nodes of the graph without containing any loop.*

**The branches of tree are called twigs or tree-branches. The number of branches or twigs, in any selected tree is always one less than the number of nodes, i.e.,**

*Twigs*Twigs = (

*n*â€“ 1), where*n*is the number of nodes of the graph.For this case, twigs = (4 â€“ 1) = 3 twigs. These are shown by solid lines in Fig. (b).

**If a graph for a network is known and a particular tree is specified,**

*Links and Co-tree***are referred to as the**

*the remaining branches***. The**

*links***is called a**

*collection of links***. So, co-tree is the complement of a tree. These are shown by dotted lines in Fig. (b).**

*co-tr**ee***(a) Circuit**

**(b) Trees and links of circuit of Fig. (a)**

*Properties of a Tree*- In a tree, there exists one and only one path between any pairs of nodes.
- Every connected graph has at least one tree.
- A tree contains all the nodes of the graph.
- There is no closed path in a tree and hence, tree is circuitless.
- The rank of a tree is (
*n*â€“ 1).