forest tree nodes

Question

I'd like to clean up my tree by drawing an edge between parent nodes. In the following, I'd like to have the 0 nodes (which are children now) adjacent to the numbered nodes (ie on the "parent" level). How might I do this? Am I missing something simple? Thanks!

\documentclass{article}
\usepackage{forest}
\begin{document}

\begin{forest}
for tree={circle, draw, l sep=20pt, s sep=5 mm}
[0, green
    [0, green]
    [1, red
        [0, green]
        [2, red
            [0, green]
            [3, red]
        ]
        [3, red
            [0, green]
            [2, red]
        ]
    ]
    [2, red
        [0, green]
        [1, red
            [0, green]
            [3, red]
        ]
        [3, red
            [0, green]
            [1, red]
        ]
    ]
    [3, red
        [0, green]
        [1, red
            [0, green]
            [2, red]        
        ]
        [2, red
            [0, green]
            [1, red]            
        ]
    ]
]
\end{forest}
\end{document}

by letting each 0 node be on the same level as the parents and connecting the parent.

That is, I'd like it to look something like

0 ----- 0
       / \
0 --- 1   2 --- 0
     / \
    2   3

Answer 1

This not really an answer but more to clarify if you want something of this sort.

\documentclass{article}
\usepackage{forest}
\begin{document}
\begin{forest}
for tree={circle, draw, l sep=20pt, s sep=15mm}
[0, green,alias=L1
%    [0, green]
    [1, red,alias=L2
%        [0, green]
        [2, red,alias=L3
%            [0, green]
            [3, red]
        ]
        [3, red,alias=L4
%            [0, green]
            [2, red]
        ]
    ]
    [2, red,alias=L5
%        [0, green]
        [1, red,alias=L6
%            [0, green]
            [3, red]
        ]
        [3, red,alias=L7
%            [0, green]
            [1, red]
        ]
    ]
    [3, red,alias=L8
%        [0, green]
        [1, red,alias=L9
%            [0, green]
            [2, red]        
        ]
        [2, red,alias=L10
%            [0, green]
            [1, red]            
        ]
    ]
]
\foreach \X in {1,...,10}
{\ifnum\X=1
\draw (L\X) -- ++ (-1.5cm,0) node[circle,left,draw,green]{0};
\else
\draw (L\X) -- ++ (-1cm,0) node[circle,left,draw,green]{0};
\fi}
\end{forest}
\end{document}

Answer 2

Not a tikz-tree solution: I just try to replicate your graph using normal TikZ

\documentclass[tikz,margin=3mm]{standalone}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\node [circle,draw] (0c) at (0,0) {0};
\node [circle,draw] (1) at ($(0c)+(-120:1.5)$) {1};
\node [circle,draw] (2r) at ($(0c)+(-60:1.5)$) {2};
\node [circle,draw] (2l) at ($(1)+(-120:1.5)$) {2};
\node [circle,draw] (3) at ($(1)+(-60:1.5)$) {3};
\node [circle,draw] (0al) at ($(0c)+(-1.5,0)$) {0};
\node [circle,draw] (0bl) at ($(1)+(-1.5,0)$) {0};
\node [circle,draw] (0r) at ($(2r)+(1.5,0)$) {0};
\draw (0c)--(1)
    (0c)--(2r)
    (1)--(2l)
    (1)--(3)
    (0c)--(0al)
    (1)--(0bl)
    (2r)--(0r);
\end{tikzpicture}
\end{document}