I am reading Emanuel Derman's paper Patterns of Volatility Change. The section, Implied Volatility In The Sticky Implied Tree Model has the linear skew approximation near the old underlying $S_0$ $$\Sigma(S,K,t)=\Sigma_0-b(K+S-2S_0)$$
A related passage is
In the linear approximation of the local volatility model you can write $\Sigma=f(S+K)$ with $\Sigma$ a function of $S+K$.
I am wondering how these are derived from the implied volatility tree model which I think is the tree version of the local volatility model. Can someone please shed light on this question?