How to convert `iptables -S` output to a breadth-first listing

I'm looking for a program that takes the output of iptables -S and converts it to a breadth-first listing. Why? I'm doing some work on a router using VyOS where several layers of tables are pre-installed, so it is difficult to trace back all the rules connect to INPUT, FORWARD, and OUTPUT. ----- As per @JeffSchaller 's [request], here is sample output that needs to be parsed:

$ sudo iptables -S 
-P INPUT ACCEPT
-P FORWARD ACCEPT
-P OUTPUT ACCEPT
-N LAN1_IN
-N MINIUPNPD
-N UBNT_FW_IN_SUSPEND_HOOK
-N UBNT_PFOR_FW_HOOK
-N UBNT_PFOR_FW_RULES
-N UBNT_VPN_IPSEC_FW_HOOK
-N UBNT_VPN_IPSEC_FW_IN_HOOK
-N VYATTA_FW_IN_HOOK
-N VYATTA_FW_LOCAL_HOOK
-N VYATTA_FW_OUT_HOOK
-N VYATTA_POST_FW_FWD_HOOK
-N VYATTA_POST_FW_IN_HOOK
-N VYATTA_POST_FW_OUT_HOOK
-N WAN_IN
-N WAN_LOCAL
-N WAN_OUT
-A INPUT -j UBNT_VPN_IPSEC_FW_HOOK
-A INPUT -j VYATTA_FW_LOCAL_HOOK
-A INPUT -j VYATTA_POST_FW_IN_HOOK
-A FORWARD -j MINIUPNPD
-A FORWARD -j UBNT_VPN_IPSEC_FW_IN_HOOK
-A FORWARD -j UBNT_PFOR_FW_HOOK
-A FORWARD -j UBNT_FW_IN_SUSPEND_HOOK
-A FORWARD -j VYATTA_FW_IN_HOOK
-A FORWARD -j VYATTA_FW_OUT_HOOK
-A FORWARD -j VYATTA_POST_FW_FWD_HOOK
-A OUTPUT -j VYATTA_POST_FW_OUT_HOOK
-A LAN1_IN -m comment --comment LAN1_IN-10 -m state --state INVALID -j LOG --log-prefix "[LAN1_IN-10-D]"
-A LAN1_IN -m comment --comment LAN1_IN-10 -m state --state INVALID -j DROP
-A LAN1_IN -p udp -m comment --comment LAN1_IN-20 -m udp --dport 53 -m set --match-set dnsaddr dst -j RETURN
-A LAN1_IN -p udp -m comment --comment LAN1_IN-30 -m set --match-set dnsaddr src -m udp --dport 53 -j RETURN
-A LAN1_IN -m comment --comment LAN1_IN-60 -m state --state NEW -j RETURN
-A LAN1_IN -m comment --comment LAN1_IN-70 -m state --state RELATED -j RETURN
-A LAN1_IN -m comment --comment LAN1_IN-80 -m state --state ESTABLISHED -j RETURN
-A LAN1_IN -m comment --comment "LAN1_IN-10000 default-action drop" -j LOG --log-prefix "[LAN1_IN-default-D]"
-A LAN1_IN -m comment --comment "LAN1_IN-10000 default-action drop" -j DROP
-A VYATTA_FW_IN_HOOK -i eth0 -j WAN_IN
-A VYATTA_FW_IN_HOOK -i eth1 -j LAN1_IN
-A VYATTA_FW_LOCAL_HOOK -i eth0 -j WAN_LOCAL
-A VYATTA_FW_OUT_HOOK -o eth0 -j WAN_OUT
-A VYATTA_POST_FW_FWD_HOOK -j ACCEPT
-A VYATTA_POST_FW_IN_HOOK -j ACCEPT
-A VYATTA_POST_FW_OUT_HOOK -j ACCEPT
-A WAN_IN -m comment --comment WAN_IN-10 -m state --state ESTABLISHED -j RETURN
-A WAN_IN -m comment --comment WAN_IN-20 -m state --state RELATED -j RETURN
-A WAN_IN -m comment --comment WAN_IN-30 -m state --state INVALID -j LOG --log-prefix "[WAN_IN-30-D]"
-A WAN_IN -m comment --comment WAN_IN-30 -m state --state INVALID -j DROP
-A WAN_IN -m comment --comment "WAN_IN-10000 default-action drop" -j DROP
-A WAN_LOCAL -m comment --comment WAN_LOCAL-10 -m state --state ESTABLISHED -j RETURN
-A WAN_LOCAL -m comment --comment WAN_LOCAL-20 -m state --state RELATED -j RETURN
-A WAN_LOCAL -m comment --comment WAN_LOCAL-30 -m state --state INVALID -j LOG --log-prefix "[WAN_LOCAL-30-D]"
-A WAN_LOCAL -m comment --comment WAN_LOCAL-30 -m state --state INVALID -j DROP
-A WAN_LOCAL -m comment --comment "WAN_LOCAL-10000 default-action drop" -j LOG --log-prefix "[WAN_LOCAL-default-D]"
-A WAN_LOCAL -m comment --comment "WAN_LOCAL-10000 default-action drop" -j DROP
-A WAN_OUT -m comment --comment WAN_OUT-10 -m state --state NEW -j RETURN
-A WAN_OUT -m comment --comment WAN_OUT-20 -m state --state RELATED -j RETURN
-A WAN_OUT -m comment --comment WAN_OUT-30 -m state --state ESTABLISHED -j RETURN
-A WAN_OUT -m comment --comment WAN_OUT-40 -m state --state INVALID -j LOG --log-prefix "[WAN_OUT-40-D]"
-A WAN_OUT -m comment --comment WAN_OUT-40 -m state --state INVALID -j DROP
-A WAN_OUT -m comment --comment "WAN_OUT-10000 default-action drop" -j LOG --log-prefix "[WAN_OUT-default-D]"
-A WAN_OUT -m comment --comment "WAN_OUT-10000 default-action drop" -j DROP

-------------- I am selecting @LL3 's answer as correct, first past the post. @LL3 's answer has since been modified to be able to read stdin so I remove the patch doing same

Kudos to perl-master @JeffSchaller 's (slightly later) complete answer showing both a breadth-first listing and separately a *graphviz* output.