Item 'layer 7' already exists in the treeview : in Maya

you are trying to put your “layer 7” into two different parents, that why

cmds.treeView( control, e=True, addItem = ("layer 7", "layer 2"))
cmds.treeView( control, e=True, addItem = ("layer 7", "layer 3"))

=== Edit ===

I’ve miss-read your question. So it is a limitation on how maya.cmds wrap Qt QTreeView.
If you want to have twice layer 7 in your graph, here is how I would do this in cmds :

import maya.cmds as cmds
from collections import defaultdict

treepop = defaultdict(list)
treepop[""]=['layer_1_0',
             'layer_2_0',
             'layer_3_0',
             'layer_4_0',
             'layer_5_0',
             'layer_6_0',
             'layer_12_0']
treepop["layer_2_0"] = ['layer_7_1',
                        'layer_9_1',
                        'layer_11_1']
treepop["layer_3_0"] = ['layer_7_2']
treepop["layer_8_0"] = ['layer_10_1']
treepop["layer_10_0"] = ['layer_13_1']


window = cmds.window()
layout = cmds.formLayout()

control = cmds.treeView( parent = layout, numberOfButtons = 0, abr = False )

cmds.formLayout(layout,e=True, attachForm=(control,'top', 2))
cmds.formLayout(layout,e=True, attachForm=(control,'left', 2))
cmds.formLayout(layout,e=True, attachForm=(control,'bottom', 2))
cmds.formLayout(layout,e=True, attachForm=(control,'right', 2))

cmds.showWindow( window )

for k in treepop[""]:
    cmds.treeView( control, e=True, addItem=(k, ""))
for k in treepop:
    if k == "":
        continue
    for v in treepop[k]:
        cmds.treeView( control, e=True, addItem=(v, k))

so you can strip the last ‘_’ to get the name of the layer.

Otherwise, wrap yourself QtWidgets.QTreeView or QTreeWidget, it might be even easier

=== EDIT 2 ===

I have an example on how you can extend maya.cmds with qtreewidget :

from PySide2 import QtWidgets
from maya import OpenMayaUI, cmds
from collections import defaultdict
import shiboken2

treepop = defaultdict(list)
treepop[""]=['layer_1',
             'layer_2',
             'layer_3',
             'layer_4',
             'layer_5',
             'layer_6',
             'layer_12']
treepop["layer_2"] = ['layer_7',
                        'layer_9',
                        'layer_11']
treepop["layer_3"] = ['layer_7']
treepop["layer_8"] = ['layer_10']
treepop["layer_10"] = ['layer_13']

def mayaToQT(name):
    # Maya -> QWidget
    ptr = OpenMayaUI.MQtUtil.findControl(name)
    if ptr is None:         ptr = OpenMayaUI.MQtUtil.findLayout(name)
    if ptr is None:         ptr = OpenMayaUI.MQtUtil.findMenuItem(name)
    if ptr is not None:     return shiboken2.wrapInstance(long(ptr),
                                                          QtWidgets.QWidget)


cmds.window()
layout = cmds.columnLayout(rowSpacing=10, columnWidth=250)
qwidget= mayaToQT(layout)
qlayout = qwidget.children()[0]
treeview = QtWidgets.QTreeWidget()
treeview.setSelectionMode(QtWidgets.QAbstractItemView.ExtendedSelection)
treeview.setColumnCount(1)
treeview.setAlternatingRowColors(True)
treeview.clear()
qlayout.addWidget(treeview)

items = []
for k in treepop[""]:
    root_item = QtWidgets.QTreeWidgetItem()
    root_item.setText(0,k)
    print(k)

    if k in treepop:
        for v in treepop[k]:
            child_item = QtWidgets.QTreeWidgetItem()
            child_item.setText(0,v)
            root_item.addChild(child_item)

    items.append(root_item)
treeview.addTopLevelItems(items)

cmds.showWindow()

def getSelected():
    items = treeview.selectedItems()
    for i in items:
        print(i.text(0))