The TDE stand for "Trinity Desktop Environment", the TDE project began as a continuation of the K Desktop Environment (KDE) version 3. The name Trinity was chosen partly because the word means "three" and TDE was a continuation of KDE 3. TDE now is its own computer desktop environment project. The TDE project was founded by and is still led by Timothy Pearson. Timothy is an experienced and skilled software developer and was the KDE 3.x coordinator of previous Kubuntu releases.
The TDE project offers desktop environment for Unix-like operating systems, the Trinity desktop environment was forked from KDE a long time ago. KDE 3.5 was great but after some modifications by developers that didn't made please KDE fans, so TDE preserve KDE 3.5 after the developers.
The Trinity desktop environment is for people who really liked how things were in past and now how things are evolving now. So if you were a fan of KDE 3.5 and you missed it, here it's for you again. It can run on any supported version of major Linux distributions such as Debian, Ubuntu, Redhat, CentOS, Fedora, Mageia, OpenSUSE, and PCLinuxOS. If you don't want to install it on your distribution, there is even a Live-CD available.

TDE is a Flexible and highly customizable; Visually appealing; Equipped with a suitable collection of desktop effects that remain compatible with older hardware; Responsive on older hardware, while also being compatible with newer hardware; Compatible with, but not requiring, 3D acceleration; Compatible with freedesktop.org and Linux file system hierarchy standards; Compatible with other desktop environments; Capable of offering maximal HCI (Human-Computer Interaction) efficiency in a mouse/keyboard environment; Traditional panel, task bar, task manager, quick launch; Several text editors, file manager, image viewers, office apps, archive manager; Monitor and Display control center module for system wide single/multi monitor and display configuration; SmartCard support; Supports GTK2/Qt theme engine; tabs, check boxes, menu backgrounds; A built-in X11 compositor; A TDE-specific DBUS notification client for improved integration with common applications such as Firefox and NetworkManager (not dependant on HAL); TQt interface (lays groundwork for selective Qt4 compatibility); Flash plugin support in Konqueror; A repository of TDE compatible software applications, including several text editors, a file manager, image viewers, office apps, an archive manager, and the original and improved classic Amarok music player. You can check full list of applications.

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Note: I've tested TDE desktop environment on Ubuntu 15.04 Vivid (Unity and Gnome were already installed), after installation logged into TDE desktop successfully but unable to login in Unity/Gnome Shell.
Also tested on Debain 7 (Gnome was already installed), and unlike Ubuntu all desktops are working fine. So install it on your own, TDE official site says it is compatible with all major desktops.

To install Trinity Desktop Environment in any Ubuntu version open Terminal (Press CTRL+ALT+T) and copy the following commands in the Terminal:

To install Trinity Desktop Environment in any Debian 8/7/6 open Terminal (Press CTRL+ALT+T) and copy the following commands in the Terminal:

TDE installation packages are also available for RedHat/CentOS, Fedora, Mageia, OpenSUSE, and PCLinuxOS. You can even download ready to use LiveCD.

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