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 $gamma$.
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