{3,1,2}

In[]:=
ListLinePlot[​​SortBy[#,First]&/@ReplaceAll[Map[VertexList,​​DecomposeGraphByTags[RecursiveFunctionCallGraph[Apply[Sequence,​​$ParametricFunctions["PlusTimesShift"]@@#],500​​],2]],F.[x_]y_:>{x,y}],Mesh->All]&[{3,1,2}]
Out[]=
In[]:=
ListPlot[​​SortBy[#,First]&/@ReplaceAll[Map[VertexList,​​DecomposeGraphByTags[RecursiveFunctionCallGraph[Apply[Sequence,​​$ParametricFunctions["PlusTimesShift"]@@#],Range[20]​​],2]],F.[x_]y_:>{x,y}],Mesh->All,Filling->Axis]&[{3,1,2}]
Out[]=
In[]:=
RecursiveFunctionCallGraph[Apply[Sequence,​​$ParametricFunctions["PlusTimesShift"]@@{3,1,2}],Range[20]​​]
Out[]=
In[]:=
DecomposeGraphByTags[RecursiveFunctionCallGraph[Apply[Sequence,​​$ParametricFunctions["PlusTimesShift"]@@{3,1,2}],Range[20]​​]]
Out[]=
1
,
,2
,
,
,

In[]:=
ListLinePlot[​​SortBy[#,First]&/@ReplaceAll[Map[VertexList,​​DecomposeGraphByTags[RecursiveFunctionCallGraph[Apply[Sequence,​​$ParametricFunctions["PlusTimesShift"]@@#],5000​​],2]],F.[x_]y_:>{x,y}]]&[{3,1,2}]
Out[]=
In[]:=
Last[DecomposeGraphByTags[RecursiveFunctionCallGraph[Apply[Sequence,​​$ParametricFunctions["PlusTimesShift"]@@{3,1,2}],Range[300]​​],2]]
This is part of the infinite tree:

What about initial conditions at different points?

{1,1,2}

[ When does the function exist ? ]

{3,1,1} TimesTimesShift ××-

[ which negative values does this reach? ]

f[n] = n - f[f[n - 2]]

1 case....

2 case ....

These are for the inner f[ ]

f[n] = n - f[f[n - 3]]