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enter image description here
As shown in the figure, the path can be generated by the following code

n=5;
Select[Tuples[{1,2,3},n],#[[1]]==1&&#[[-1]]!=1&&AllTrue[Partition[#,2,1],Unequal@@#&]&]

{{1,2,1,2,3},{1,2,1,3,2},{1,2,3,1,2},{1,2,3,1,3},{1,2,3,2,3},{1,3,1,2,3},{1,3,1,3,2},{1,3,2,1,2},{1,3,2,1,3},{1,3,2,3,2}}

I also know how to draw a binary tree

CompleteKaryTree[n, 2, VertexLabels -> {x_ :> Placed[F[x], Center]}, VertexSize -> 0]

enter image description here
But i don't know how to combine them together.

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I would suggest something like this:

n = 5;
paths = Select[Tuples[{1, 2, 3}, n], #[[1]] == 1 && #[[-1]] != 1 && AllTrue[Partition[#, 2, 1], Unequal @@ # &] &]
Graph[
 DeleteDuplicates@Catenate[
   Rule @@@ Partition[Rest@FoldList[Append, {}, #], 2, 1] & /@ paths
   ],
 VertexLabels -> {___, i_} -> i
 ]

enter image description here

This works by effectively labeling each vertex with the path needed to get to it (to avoid confusing the different vertices with the same state). So e.g. the top 1 is {1} and the 3 below that is {1,3}. This is a bit easier to see when using VertexLabels->Automatic:

enter image description here

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pathsToGraph = Graph[
    GraphUnion @@ (PathGraph[Extract[#, List /@ Range[Range @ Length @ #]]] & /@ #), 
    ##2 (* graph options *)] &;

Example:

n = 5;
paths = Select[Tuples[{1, 2, 3}, n],
  #[[1]] == 1 && #[[-1]] != 1 && AllTrue[Partition[#, 2, 1], Unequal @@ # &] &];

pathsToGraph[paths, 
  VertexSize -> Large, 
  VertexLabels -> {v_ :> Placed[Last[v], Center]}, 
  VertexStyle -> White]

enter image description here

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