Specifying Process Tree for Docker Daemon to Use

Is there any way to have the process launched by my docker run command launch under the same process tree as the program executing the docker run command?

No, all the processes managed by the Docker Engine are children of the Docker Engine (daemon). For example:

docker run supertest2014/nyan
(another shell on the same machine)
ps faux
...
root       762  2.5  3.3 462068 16676 ?        Ssl  19:07   0:10 /usr/bin/docker -d
root      1563  0.0  0.1   4444   648 ?        Ss   19:13   0:00  \_ /bin/sh -c nyancat
root      1573  0.2  0.0   4228   352 ?        S    19:13   0:00      \_ nyancat
...

There you can see the docker daemon running (docker -d) and the CMD in the nyan container was /bin/sh -c nyancat which then launched nyancat.

You can however have your container share the process ID space and IPC namespace with another process, even the host. For instance, if you normally run a container, it can only see the processes running inside the container:

root@processtree:~# docker run -it --name containerpid supertest2014/nyan
<another window>
root@processtree:~# docker exec -it containerpid /bin/bash                                                                                                                                        
<note this is the WHOLE process tree>
root@355f839c31bc:/# ps faux
USER       PID %CPU %MEM    VSZ   RSS TTY      STAT START   TIME COMMAND
root        11  0.5  0.3  18132  1840 ?        Ss   23:25   0:00 /bin/bash
root        19  0.0  0.2  15564  1092 ?        R+   23:25   0:00  \_ ps faux
root         1  0.0  0.1   4444   648 ?        Ss   23:13   0:00 /bin/sh -c nyancat
root        10  0.2  0.0   4228   352 ?        S    23:13   0:01 nyancat

But if you run with --pid host then the process tree shows everything on the host:

...
root@processtree:~# docker run -it --name hostpid --pid host supertest2014/nyan
<another window>
root@processtree:~# docker exec -it hostpid /bin/bash
<note this is a partial process tree, there was a lot more going on!>
root@processtree:~# ps faux
root         1  0.0  0.5  33500  2820 ?        Ss   23:07   0:01 /sbin/init
root       356  0.0  0.1  19472   648 ?        S    23:07   0:00 upstart-udev-bridge --daemon
root       361  0.0  0.3  51360  1632 ?        Ss   23:07   0:00 /lib/systemd/systemd-udevd --daemon
root       503  0.0  0.1  15256   652 ?        S    23:07   0:00 upstart-socket-bridge --daemon
102        721  0.0  0.2  39224  1256 ?        Ss   23:07   0:00 dbus-daemon --system --fork
root       762  1.4  3.6 700744 18156 ?        Ssl  23:07   0:19 /usr/bin/docker -d
root      1563  0.0  0.1   4444   648 ?        Ss   23:13   0:00  \_ /bin/sh -c nyancat
root      1573  0.2  0.0   4228   352 ?        S    23:13   0:02  |   \_ nyancat
root      2040  0.0  0.1   4444   652 ?        Ss+  23:23   0:00  \_ /bin/sh -c nyancat
root      2048  0.2  0.0   4228   360 ?        S+   23:23   0:00  |   \_ nyancat
root      2183  0.5  0.3  18152  1848 ?        Ss   23:28   0:00  \_ /bin/bash
root      2190  0.0  0.2  15720  1216 ?        R+   23:28   0:00      \_ ps faux
root       819  0.0  0.3  43448  1756 ?        Ss   23:07   0:00 /lib/systemd/systemd-logind
syslog     833  0.0  0.3 255840  1612 ?        Ssl  23:07   0:00 rsyslogd
root       863  0.0  0.1  15272   628 ?        S    23:07   0:00 upstart-file-bridge --daemon
root       937  0.0  0.1  15816   956 ?        Ss+  23:07   0:00 /sbin/getty -8 38400 tty4
root       941  0.0  0.1  15816   952 ?        Ss+  23:07   0:00 /sbin/getty -8 38400 tty5
root       950  0.0  0.1  15816   960 ?        Ss+  23:07   0:00 /sbin/getty -8 38400 tty2
root       951  0.0  0.1  15816   960 ?        Ss+  23:07   0:00 /sbin/getty -8 38400 tty3
root       957  0.0  0.1  15816   956 ?        Ss+  23:07   0:00 /sbin/getty -8 38400 tty6
root       987  0.0  0.6  61364  3068 ?        Ss   23:07   0:00 /usr/sbin/sshd -D
...