Abstract
It is shown that “conservative” trees, which posses some measure that is conserved across branchpoints, generate a distribution of the measure over the segments that is approximately hyperbolic. When the trees become large, the shape of the distribution becomes independent of drastic changes in the asymmetry of the branching, in the measure distribution of the root segments, or in the valency of the branchpoints. In contrast to the invariant shape, the amplitude of the distribution does depend on the above parameters, but it can be computed numerically or, in many important cases, analytically. These results can be used to compute the distribution of related (but nonconserved) quantities of the trees, to solve the scaling problem of large trees, and to gain insight into the relations between local and global consequences of dendritic growth.
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