Is it possible to check if a connection is local or via ssh *after* elevating to the root account with `sudo`?

I want to see if I'm connected via ssh or running a local terminal.

If I just ssh into a server without changing to root via sudo, it's easy. Any of the variables $SSH_CLIENT, $SSH_CONNECTION or $SSH_TTY can be used to check if one is connected via SSH or local.

Problem: When I elevate to the root account with sudo -i to do administrative stuff, neither of these variables are helpful - they are all empty.

What is the best way to find out if the connection is local or via SSH then?

EDIT: With the accepted answer, it is easy to have an unobtrusive bash prompt which reflects ssh status and privileges:

if [ "$color_prompt" = yes ]; then
    # when system is accessed via SSH, hostname with light grey background
    if [[ $(pstree -s $$) = *sshd* ]]; then sshbg="\[\033[48;5;7m\]"; fi
    # when used as root, change username to orange and '#' to red for prompt
    if [ $(id -u) -eq 0 ]; then usercol="\[\033[38;5;3m\]"; hashcol="\[\033[38;5;1m\]"; else usercol="\[\033[38;5;2m\]"; fi
    # bash PS1 prompt
    PS1="${usercol}\u\[$(tput sgr0)\]@\[$(tput sgr0)\]\[\033[38;5;4m\]${sshbg}\h\[$(tput sgr0)\]:\[$(tput sgr0)\]\[\033[38;5;6m\]\w\[$(tput sgr0)\]${hashcol}\\$ \[$(tput sgr0)\]"
    unset sshbg rootcol hashcol
fi

The timed version of the pstree part runs in less than 20ms, so it can be used without introducing noticeable delays.