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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.
Twigs 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 = (n – 1), where n is the number of nodes of the graph.
For this case, twigs = (4 – 1) = 3 twigs. These are shown by solid lines in Fig. (b).
Links and Co-tree If a graph for a network is known and a particular tree is specified, the remaining branches are referred to as the links. The collection of links is called a co-tree. So, co-tree is the complement of a tree. These are shown by dotted lines in Fig. (b).

(a) Circuit

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

The branches of a co-tree may or may not be connected, whereas the branches of a tree are always connected.

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

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