The CODC believes the robinia tree was attacked on Friday night, and the tree was subsequently removed due to a safety risk.
It was reported to council by a contractor the following day and has since been reported to police.
In a statement, the CODC said it was "really disappointed to discover that someone has taken it upon themselves to attack this tree in Alexandra Cemetery recently".
"For public safety, we’ve now had to fell it."
The council also confirmed damage to speed signs on either side of the road into the cemetery. It is unclear if the incidents are related.
A council spokesperson said to the CODC's knowledge, there had been no prior complaints or issues with the tree.
Council parks and recreation manager Gordon Bailey said there would be a replanting. Wood from the felled tree would be given to the Salvation Army for firewood.
The cost to ratepayers for removal, clean-up and replanting was estimated at $1000 to $2000, Mr Bailey said.
Community members have taken to social media to express their sadness at the loss of the tree.
One person commented: "We visited Alexandra cemetery on Sunday and we both commented how peaceful it was, so it's really sad to hear this has happened."
Another said: "It was one of the few trees in the cemetery to give shelter from the sun, by a seat, for grieving people to find some solace."
Others expressed anger and many hoped there would be punishment for the perpetrator or perpetrators.
"What is wrong with people?" one Facebook user wrote.
CODC is reminding residents that cemeteries are places that should be treated with respect.
"A cemetery should be respected as a sacred resting place for those who are no longer with us," the statement said.
"If you have concerns about any trees on council-maintained land, let us know. Please don't try to cut them down or trim them yourself."
Anyone with information is urged to contact the council or police.
