Symbolic network analysis with the valid trees and the valid tree-pairs

The valid tree and the valid tree-pair as well as their values are defined. Algorithms to look for all valid trees and valid tree-pairs and to obtain their values are given. A theorem points out that the determinant of a closed network is equal to the sum of all valid tree's values and all valid tree-pair's values of the network. A unified topological formula for general active circuits to solve various types of symbolic network functions is put forward.