Visualizing Cayley's Formula in Mathematica

You can use Prüfer sequences to generate all labelled trees. IGraph/M has the required functionality.

coloredPruferTree[p_] := 
  IGFromPrufer[p, GraphStyle -> "BasicBlack", VertexSize -> 1/4] // 
   IGVertexMap[ColorData[97], VertexStyle -> VertexList]

allTrees[n_] := 
  coloredPruferTree /@ Tuples[Range /@ ConstantArray[n, n - 2]]

Now you can list all labelled trees on 3 vertices:

allTrees[3]

Or 4 vertices:

allTrees[4]

Furthermore, you can group the trees by isomorphism class using

allTrees[5] // GroupBy[CanonicalGraph] // Values

I have upvoted @Szabolcs answer. I am sure there are better code to tree approaches than mine in the following. IGraphM is ideal as Szabolcs illustrates. I post this just to extend visualization for $n>4$ case.

Happy New Year to all MSE users

Just some visualization options:

fun[code_] := 
 Module[{v = Range[Length[code] + 2], cd = code, e = {}, c},
  While[Length[v] != 2,
   c = Sort[Complement[v, cd]];
   AppendTo[e, {cd[[1]], c[[1]]}];
   v = DeleteCases[v, c[[1]]];
   cd = Drop[cd, 1];];
  Graph[UndirectedEdge @@@ AppendTo[e, v], VertexSize -> 0.3, 
   VertexLabels -> 
    Table[i -> Placed[Style[i, White, Bold], {1/2, 1/2}], {i, 
      v[[-1]]}], 
   VertexStyle -> 
    Table[i -> ColorData["Rainbow"][i/v[[-1]]], {i, v[[-1]]}]]]
disp[n_] := 
 Grid[Partition[Column[{#, fun[#]}] & /@ Tuples[Range[n], n - 2], n], 
  Frame -> All]
man[u_] := 
 Manipulate[fun[{##}[[All, 1]]], ##, ControlType -> SetterBar] & @@ 
  Table[{Symbol["x" <> ToString[i]], Range[u]}, {i, u - 2}]

So, disp[4]:

And man[10] (note this is inspired by this Wolfram Demonstration but generalizes the visualization and I do not use the same code for creating tree from Prufer code). :