The kingdom of God is the messianic kingdom that the Old Testament predicted. It would be an earthly kingdom over which Messiah would rule for 1, 000 years (Rev. 20:4-6). It is similar to a mustard seed in that it had a small beginning in the preaching of Jesus, but it will grow to be a very large entity. It will eventually encompass the whole earth and the entire human race (Ps. 2; et al.).
The reference to the birds nesting in its branches may simply be an insignificant detail. However it is probably an allusion to the tree in Nebuchadnezzar's dream in which the birds evidently represent the Gentile nations that profit from the tree (kingdom, Dan. 4:7-23). Several Old Testament passages use a tree with birds flocking to its branches to illustrate a kingdom that people perceive as great (Judg. 9:15; Ps. 104:12-13; Ezek. 17:22-24; 31:3-14).
The point of the parable is the growth of the kingdom from a very small and apparently insignificant beginning to a very large entity. Particularly its small beginning is in view. In this context Luke probably wanted his readers to connect the great power of Jesus manifested in the woman's healing (vv. 10-17) and the power that results in the tree's unusual growth.