Are we supposed to install busybox after rooting?

You don't need busybox unless some of your apps require it

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BusyBox combines tiny versions of many common UNIX utilities into a single small executable. It provides replacements for most of the utilities you usually find in GNU fileutils, shellutils, etc. The utilities in BusyBox generally have fewer options than their full-featured GNU cousins; however, the options that are included provide the expected functionality and behave very much like their GNU counterparts.

If you have a root app that need BusyBox to function then yes you need it. Most root apps I have on my phone have their own counterpart already imbedded in the apps, for example Titanium Backup, which is why it works without BusyBox.

Also most custom roms already have busybox included as standard. BusyBox might be needed on Stock, only if you have a root app that needs it to function.

Btw it wont hurt you to install it if you are unsure if any of your app requires it

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