How and why do these tree-like river structures form?
The river isn't splitting. The picture shows a number of dendritic drainage channels flowing into a larger river/stream. They are the most common form of topographical draining system. They develop in gently sloping topography and they ...


  develops in regions underlain by homogeneous material. That is, the subsurface geology has a similar resistance to weathering so there is no apparent control over the direction the tributaries take. Tributaries joining larger streams at acute angle (less than 90 degrees) [Earth Drainage Systems]. 


You'll notice in the picture all the drainage channels intersect the river at approximately 90 degrees. Also it appears the channels dendritic channels have formed in what looks like a sandy/silty sediment, again homogenous material.

Other sources that may be helpful:

Experiment to Examine Dendritic River Systems

Dendritic river flow simulations

Flow direction &amp; branching geometry

Dendritic Design