Which is the best file explorer for tree style folder browsing in Gnome under Ubuntu 18.04? Earlier, file explorer under Unity in 16.04 used thi do it, but not in 18.04.
Files, aka nautilus, has a feature where folders can be expanded in a tree view style in the main pane. Under "Preferences", check "Allow folders to be expanded" on the View tab.
If you are specifically looking for the traditional navigation tree in the left pane, you will need to move to another file manager, because nautilus dropped that feature some time ago. Both Thunar and PCManFM are excellent light-weight file managers that are GTK based, so fit well within a gnome environment, and that support tree view in the left pane. Alternatively, you can try forks of nautilus that maintained that feature, i.e. nemo, an early fork of nautilus 3, or caja, the continuation of nautilus 2. Which is the best? Very personal, so you probably should try them and decide for yourself.
It can be done in "Files" app, which is usually preinstalled. The trick is to hint Preferences under the Hamburger menu and than check a checkbox "Allow folders to be expanded".