How to show the tree view in left pane of Nautilus?

This is the most near that you can get of what you want, otherwise look for another file manager:

Look for 'Preferences', in the 'Views' tab, select "View new folders using: list view", then select the 'Display' tab, there will be a Checkbox that tells you 'Navigate folders in a tree', close Nautilus. Now open a folder and select list view, that way you will have a tree like behavior in the main panel.

If you don't like how it's shown then try with another file manager.

Nautilus 3.5.4 and later

William Jon McCann removed this feature in version 3.5.4:

Use a list model instead of a tree model

It is the list view after all. Tree models don't work well on touch and it isn't consistent with the file chooser.

Nautilus 3.4.2

Select menu View then menu item Sidebar and then select Tree.

Toggle the sidebar between hidden/shown with F9

On gnome-shell 3.28 (Ubuntu 18.04) this command works:

gsettings set org.gnome.nautilus.list-view use-tree-view true