a tree is similarly the height of the largest complete binary tree that can be embedded below that node. Any node with Strahler number i must have at least two descendants with Strahler number i − 1, at least four descendants with Strahler number i − 2, etc., and at least 2i − 1 leaf descendants. Therefore, in a tree with n nodes, the largest possible Strahler number is log2 n + 1. However, unless the tree forms a complete binary tree its Strahler number will be less than this bound. In an n-node binary tree, chosen uniformly at random among all possible binary trees, the expected index of the root is with high probability very close to log4 n. Applications River networks In the application of the Strahler stream order to hydrology, each segment of a stream or river within a river network is treated as a node in a tree, with the next segment downstream as its parent. When two first-order streams come together, they form a second-order stream. When two second-order streams come together, they form a third-order stream. Streams of lower order joining a higher order stream do not change the order of the higher stream. Thus, if a first-order stream joins a second-order stream, it remains a second-order stream. It is not until a second-order stream combines with another second-order stream that it becomes a third-order stream. As with mathematical trees, a segment with index i must be fed by at least 2i − 1 different tributaries of index 1. Shreve noted that Horton’s and Strahler’s Laws should be expected from any topologically random distribution. A later review of the relationships confirmed this argument, establishing that, from the properties the laws describe, no conclusion can be drawn to explain the structure or origin of the stream network. To qualify as a stream a hydrological feature must be either recurring or perennial. Recurring (or "intermittent") streams have water in the channel for at least part of the year. The index of a stream or river may range from 1 (a stream with no tributaries) to 12 (globally the most powerful river, the Amazon, at its mouth). The Ohio River is of order eight and the Mississippi River is of order 10. Estimates are that 80% of the streams on the planet are first to third order headwater streams. If the bifurcation ratio of a river network