This paper investigates the multiscale statistical structure of the area and width functions of simulated and real river networks via state-of-the-art wavelet-based multifractal (MF) formalisms. First, several intricacies in performing MF analysis of these signals are discussed, and a robust framework for accurate estimation of the MF spectrum is presented. Second, it considers the following three questions: (1) Does the topology of river networks leave a unique signature on the MF spectrum of area and width functions? (2) How different are the MF properties of commonly used simulated trees and those of real river networks? and (3) Are there differences between the MF properties of width and area functions, and what can these tell us about the topology of hillslope versus channelized drainage patterns in a river basin? The results indicate discrepancies between the statistical scaling of the area functions of real networks (found to be multifractal with a considerable spread of local singularities and the most prevailing singularity ranging from 0.4 to 0.8) and that of several commonly used stochastic self-similar networks (found to be monofractal with a single singularity exponent H in the range of 0.5-0.65). Moreover, differences are found between the MF properties of width and area functions of the same basin. These differences may be the result of distinctly different branching topologies in the hillslope versus channelized drainage paths and need to be further investigated.