  Various real-life networks exhibit degree correlations and heterogeneous
structure, with the latter being characterized by power-law degree distribution
$P(k)\sim k^{-\gamma}$, where the degree exponent $\gamma$ describes the extent
of heterogeneity. In this paper, we study analytically the average path length
(APL) of and random walks (RWs) on a family of deterministic networks,
recursive scale-free trees (RSFTs), with negative degree correlations and
various $\gamma \in (2,1+\frac{\ln 3}{\ln 2}]$, with an aim to explore the
impacts of structure heterogeneity on APL and RWs. We show that the degree
exponent $\gamma$ has no effect on APL $d$ of RSFTs: In the full range of
$\gamma$, $d$ behaves as a logarithmic scaling with the number of network nodes
$N$ (i.e. $d \sim \ln N$), which is in sharp contrast to the well-known double
logarithmic scaling ($d \sim \ln \ln N$) previously obtained for uncorrelated
scale-free networks with $2 \leq \gamma <3$. In addition, we present that some
scaling efficiency exponents of random walks are reliant on degree exponent
$\gamma$.
