Nested list to graph

The following nested list can be regarded as a representation of a (tree) graph:

li = {"fig", {"date", {"kumquat"}, {"papaya", {"peach"}, {"apple"}}},
             {"mango", {"orange", {"pear"}, {"avocado"}}}, 
             {"banana"}}

In the above, a string is a node in the tree, and any lists that follow it are subtrees rooted at that node.

What are some of the ways by which this can be converted into a graph (or more concretely, a list of DirectedEdges)? I've come up with one way, listed below. But I wanted to learn about other interesting approaches - for instance, pattern replacements might be used?

This is what I came up with:

h[{str_String}] := Sequence[];
h[{str_String, ls__List}] := {DirectedEdge[str, #[[1]]], h@#} & /@ {ls};

edges = Flatten@h@li

(* 
{"fig" \[DirectedEdge] "date", "date" \[DirectedEdge] "kumquat", 

 "date" \[DirectedEdge] "papaya", "papaya" \[DirectedEdge] "peach", 
 "papaya" \[DirectedEdge] "apple", "fig" \[DirectedEdge] "mango", 
 "mango" \[DirectedEdge] "orange", "orange" \[DirectedEdge] "pear", 
 "orange" \[DirectedEdge] "avocado", "fig" \[DirectedEdge] "banana"}
*)

TreePlot[Rule @@@ edges, Automatic, "fig", DirectedEdges -> True, 
 VertexLabeling -> True]

Answer

edges = Cases[li,
   {node_String, subtrees__List} :> (
     node \[DirectedEdge] #[[1]] & /@ {subtrees}),
   {0, ∞}] // Flatten

Note the level specification within Cases.

Graph[edges,
 VertexLabels -> "Name",
 ImagePadding -> 30,

 GraphLayout -> {
   "LayeredEmbedding",
    "RootVertex" -> "fig"}]