Formulation and asymptotic properties of the bifurcation ratio in Horton’s law for the equiprobable binary tree model

The bifurcation ratio for the equiprobable binary tree model is formulated. We obtain the exact expression of the $k\text{th}$ moment of the second-order streams. We also obtain a recursive equation between $r\text{th}$ and $(r+1)\text{th}$ order streams. Horton’s law is confirmed numerically by calculating this recursive equation and asymptotic properties of the bifurcation ratio are discussed.

Figure 1

A binary tree of magnitude 3 $(n=3)$. (a) Ordinary binary tree. (b) Modified binary tree. The number on each node shows its own Horton-Strahler number.

Figure 2

Some examples of a perfect binary tree. (a) 2-perfect binary tree, (b) 3-perfect binary tree, and (c) 4-perfect binary tree.

Figure 3

Procedure to make a binary tree in the case of $n=14$, $m=5$. (a) A binary tree of magnitude 5, where △ represents leaf. (b) Replacing △ nodes with two perfect binary trees. (c) Hanging $4(=n-2m)$ nodes (gray nodes are hanging nodes). (d) Remaking the binary tree.

Figure 4

Procedure to make a binary tree in the case of $n=19$, $n_2=8$, $n_3=3$. (a) A binary tree of magnitude $3(=n_3)$, where △ represents leaf. (b) Replacing △ nodes with 3-perfect binary trees. (c) Hanging $2(=n_2-2n_3)$ nodes. (d) Replacing hanging nodes with 2-perfect binary trees. (e) Hanging $3(=n-2n_2)$ nodes. (f) Remaking the binary tree.

Figure 5

$k\text{th}$ moment ratio of $S_{r,n}$ to $S_{r+1,n}$ for $k=1$ (a), $k=2$ (b), $k=3$ (c), and $k=4$ (d).

Figure 6

Log-log plot of $R\left(n\right)$ for $r=1$, 2, 3, and 4. Data points are constructed from Fig. 5a, and thinned out randomly. The dashed line indicates $R\left(n\right)=\frac{2}{n}$.