Forest package: How to use forestset in the preamble?

In forestset you need to define style which name is considered in forest environment. See MWE below:

documentclass[a4paper]{article}
usepackage{forest}
forestset{my tree/.style = {% NEW
    for tree={
        inner sep=0.075cm, 
        l sep=1mm,% CHANGED AND CORRECTED
        s sep=1mm,% changed
        grow'=0,
        child anchor=west,parent anchor=south,
        anchor=west,calign=first,
        edge path={
          noexpandpath [draw, forestoption{edge}]
          (!u.south west) +(7.5pt,0) |- node{} (.child anchor)forestoption{edge label};
        },
        before typesetting nodes={
          if n=1
            {insert before={[,phantom]}}
            {}
        },
        fit=band,
        before computing xy={l=15pt},
            }% end of for tree
                        }% end of my tree style
        }% end of forestset
%-------------------------------------------------------------
begin{document}
begin{forest} my tree
 [$B$
  [$B_{1}$]
  [$B_{2}$
    [$A$
      [$A_{1}$,tikz={node [draw,inner sep=0.5pt,fit to=tree]{};}]
      [$A_{2}$
        [$D$]
        [$E$]
      ]
      [$A_{3}$
      ]
    ]
  ]
]
end{forest}
end{document}

You define the style sn edges in the preamble, but you don't say that you want your tree to use it. To do this you should provide the style name as a key in your tree.

MWE:

documentclass{article}
usepackage{forest}
forestset{ 
  sn edges/.style={
  for tree={
    grow'=0, 
    l=0pt, inner sep=0.075cm, s sep=0.1mm,
    child anchor=west,parent anchor=south,
    anchor=west,calign=first,
    edge path={
      noexpandpath [draw, forestoption{edge}]
      (!u.south west) +(7.5pt,0) |- node{} (.child anchor)forestoption{edge label};
    },
    before typesetting nodes={
      if n=1
        {insert before={[,phantom]}}
        {}
    },
    fit=band,
    before computing xy={l=15pt},
  }
 }
}
%-------------------------------------------------------------
begin{document}
begin{forest}
sn edges,
[$B$
  [$B_{1}$]
  [$B_{2}$
    [$A$
      [$A_{1}$,tikz={node [draw,inner sep=0.5pt,fit to=tree]{};}]
      [$A_{2}$
        [$D$]
        [$E$]
      ]
      [$A_{3}$
      ]
    ]
  ]
]
end{forest}
end{document}

Result:

An alternative is not to create a new style but to apply the given style automatically to all trees using the key default preamble in forestset. In this case you don't need to provide a style key to individual trees.

documentclass{article}
usepackage{forest}
forestset{ 
  default preamble={
  for tree={
    grow'=0, 
    l=0pt, inner sep=0.075cm, s sep=0.1mm,
    child anchor=west,parent anchor=south,
    anchor=west,calign=first,
    edge path={
      noexpandpath [draw, forestoption{edge}]
      (!u.south west) +(7.5pt,0) |- node{} (.child anchor)forestoption{edge label};
    },
    before typesetting nodes={
      if n=1
        {insert before={[,phantom]}}
        {}
    },
    fit=band,
    before computing xy={l=15pt},
  }
 }
}
%-------------------------------------------------------------
begin{document}
begin{forest}
[$B$
  [$B_{1}$]
  [$B_{2}$
    [$A$
      [$A_{1}$,tikz={node [draw,inner sep=0.5pt,fit to=tree]{};}]
      [$A_{2}$
        [$D$]
        [$E$]
      ]
      [$A_{3}$
      ]
    ]
  ]
]
end{forest}
end{document}