Compute the height of a radix tree
JavaScript (Firefox 30-57), 137 bytes

f=(a,b=["",...a],s=new Set(b.map(w=&gt;w[0])))=&gt;b.length&gt;1?(s.size&gt;1)+Math.max(...(for(c of s)f(a,[for(w of b)if(w&amp;&amp;w[0]==c)w.slice(1)]))):1


There must be something wrong with my algorithm as I seem to have to special-case an empty string parameter.